Question: Vanessa works out for $\frac{3}{4}$ of an hour every day. To keep her exercise routines interesting, she includes different types of exercises, such as jumping jacks and push-ups, in each workout. If each type of exercise takes $\frac{3}{8}$ of an hour, how many different types of exercise can Vanessa do in each workout?
Solution: To find out how many types of exercise Vanessa could do in each workout, divide the total amount of exercise time ( $\frac{3}{4}$ of an hour) by the amount of time each exercise type takes ( $\frac{3}{8}$ of an hour). $ \dfrac{{\dfrac{3}{4} \text{ hour}}} {{\dfrac{3}{8} \text{ hour per exercise}}} = {\text{ number of exercises}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{3}{8} \text{ hour per exercise}}$ is ${\dfrac{8}{3} \text{ exercises per hour}}$ $ {\dfrac{3}{4}\text{ hour}} \times {\dfrac{8}{3} \text{ exercises per hour}} = {\text{ number of exercises}} $ $ \dfrac{{3} \cdot {8}} {{4} \cdot {3}} = {\text{ number of exercises}} $ Reduce terms with common factors by dividing the $3$ in the numerator and the $3$ in the denominator by $3$ $ \dfrac{{\cancel{3}^{1}} \cdot {8}} {{4} \cdot {\cancel{3}^{1}}} = {\text{ number of exercises}} $ Reduce terms with common factors by dividing the $8$ in the numerator and the $4$ in the denominator by $4$ $ \dfrac{{1} \cdot {\cancel{8}^{2}}} {{\cancel{4}^{1}} \cdot {1}} = {\text{ number of exercises}} $ Simplify: $ \dfrac{{1} \cdot {2}} {{1} \cdot {1}} = {2} $ Vanessa can do 2 different types of exercise per workout.